System and method for indexing a data stream

ABSTRACT

There are provided methods, computer program products, and systems for indexing a data stream. A method for indexing a data stream having attribute values includes the steps of parsing the data stream, and forming an index of tuples for a subset of attribute values of the data stream. The index is configured for retrieving the top-K tuples that optimize linearly weighted sums of at least some of the attribute values in the subset.

This invention was made with Government support under Contract No.:H98230-05-3-0001 awarded by the U.S. Department of Defense. TheGovernment has certain rights in this invention.

BACKGROUND

1. Technical Field

The present invention relates generally to data stream applications and,more particularly, a system and method for indexing data streams.

2. Description of the Related Art

Data stream applications are becoming increasingly popular. Many datastream applications use various linear optimization queries to retrievethe top-K tuples that maximize or minimize the linearly weighted sums ofcertain attribute values.

For example, in environmental epidemiological applications, linearmodels that incorporate, e.g., remotely sensed images, weatherinformation, and demographic information are used to predict theoutbreak of certain environmental epidemic diseases such as, e.g.,Hantavirus Pulmonary Syndrome. In oil/gas exploration applications,linear models that incorporate, e.g., drill sensor measurements andseismic information are used to guide the drilling direction. Infinancial applications, linear models that incorporate, e.g., personalcredit history, income level, and employment history are used toevaluate individual credit risks for loan approvals.

In all the above applications, data continuously streams in (e.g., fromsatellites and sensors) at a rapid rate. Users frequently pose linearoptimization queries and want answers back as soon as possible.Moreover, different individuals may pose queries that have divergentweights and K's. This is because, e.g., the “optimal” weights may varyfrom one location to another (in oil/gas exploration), the weights maybe adjusted as the model is continually trained with historical datacollected more recently (in environmental epidemiology and finance), anddifferent users may have differing preferences.

Chang et al., in “The Onion Technique: Indexing for Linear OptimizationQueries”, SIGMOD Conf. 2000, pp. 391-402 (hereinafter the “OnionTechnique Article”), the disclosure of which is incorporated byreference herein, proposed using an onion index to speed up theevaluation of linear optimization queries against a large databaserelation. An onion index organizes all tuples in the database relationinto one or more convex layers, where each convex layer is a convexhull. For each i≧1, the (i+1)^(th) convex layer is included within theith convex layer. For any linear optimization query, to find the top-Ktuples, typically no more than all the vertices of the first K outerconvex layers in the onion index are searched.

However, due to the extremely high cost of computing precise convexhulls, both the creation and the maintenance of the onion index arerather expensive. Moreover, an onion index keeps track of all tuples ina relation and, thus, requires a lot of storage space. In a datastreaming environment, tuples keep arriving rapidly while availablememory is limited. As a result, it is very difficult to maintain aprecise onion index for a data stream, let alone using the precise onionindex to provide exact answers to linear optimization queries againstthe stream.

A description will now be given of the traditional onion index, asdisclosed in the above-referenced “Onion Technique Article”, for linearoptimization queries against a large database relation.

Suppose each tuple includes n≧1 numerical feature attributes and m≧0other non-feature attributes. A top-K linear optimization query asks forthe top-K tuples that maximize the following linear equation:

${\max\limits_{{top}\mspace{14mu} K}\{ {\sum\limits_{i = 1}^{n}{w_{i}a_{i}^{j}}} \}},$where (a₁ ^(j), a₂ ^(j), . . . , a_(n) ^(j)) is the feature attributevector of the jth tuple and (w₁, w₂, . . . , w_(n)) is the weightingvector of the query. Some w_(i)'s may be zero. Here,

$v_{j} = {\sum\limits_{i = 1}^{n}{w_{i}a_{i}^{j}}}$is called the linear combination value of the jth tuple. It is to benoted that a linear optimization query may alternatively ask for the Kminimal linear combination values. In this case, we can turn such aquery into a maximization query by switching the signs of the weights.For purposes of brevity and illustration, maximization queries areprimarily described herein after.

A set of tuples S can be mapped to a set of points in an n-dimensionalspace according to their feature attribute vectors. For a top-K linearoptimization query, the top-K tuples are those K tuples with the largestprojection values along the query direction.

Linear programming theory has the following theorem, designated hereinas Theorem 1.

Theorem 1: Given a linear maximization criterion and a set of tuples S,the maximum linear combination value is achieved at one or more verticesof the convex hull of S.

Utilizing this property, the onion index in the above-referenced “OnionTechnique Article” organizes all tuples into one or more convex layers.The first convex layer L₁ is the convex hull of all tuples in S. Thevertices of L₁ form a set S₁ ⊂S. For each i>1, the ith convex layerL_(i) is the convex hull of all tuples in

$S - {\bigcup\limits_{j = 1}^{i - 1}{S_{j}.}}$The vertices of L_(i) form a set

$S_{i} \subseteq {S - {\bigcup\limits_{j = 1}^{i - 1}{S_{j}.}}}$It is easy to see that for each i≧1, L_(i+1) is contained within L_(i).FIG. 1 illustrates an exemplary onion index 100 in two-dimensionalspace, in accordance with the prior art. The exemplary onion index 100shown in FIG. 1 includes a first convex layer 110, a second convex layer120, and a third convex layer 130.

From Theorem 1, we know that the maximum linear combination value ateach L_(i) (i≧1) is larger than all linear combination values fromL_(i)'s inner layers. Also, there may be multiple tuples on L_(i) whoselinear combination values are larger than the maximum linear combinationvalue of L_(i+1). As a result, we have the following property,designated herein as Property 1.

Property 1: For any linear optimization query, suppose all tuples aresorted in descending order of their linear combination values (v_(j)).The tuple that is ranked kth in the sorted list is called the kthlargest tuple. Then the largest tuple is on L₁. The second largest tupleis on either L₁ or L₂. In general, for any i≧1, the ith largest tuple ison one of the first i outer convex layers.

Given a top-K linear optimization query, the search procedure of theonion index starts from L₁ and searches the convex layers one by one. Oneach convex layer, all its vertices are checked. Based on Property 1,the search procedure can find the top-K tuples by searching no more thanthe first K outer convex layers.

During a tuple insertion or deletion, one or more convex layers may needto be reconstructed in order to maintain the onion index. The detailedonion index maintenance procedure is disclosed in the above-referenced“Onion Technique Article”. Both the creation and the maintenance of theonion index require computing convex hulls. This is fairly expensive, asgiven N points in an n-dimensional space, the worst-case computationalcomplexity of constructing the convex hull is O(N 1n N+N^(└n/2┘)).

It is to be noted that in some data stream applications, the linearoptimization queries are known in advance and the entire history of thestream is considered. In this case, for each linear optimization query,an in-memory materialized view can be maintained to continuously keeptrack of the top-K tuples. However, if there are many such linearoptimization queries, it may not be feasible and/or otherwise possibleto keep all these materialized views in memory and/or to maintain themin real time.

It is to be further noted that in a data streaming environment, tuplesmay continuously arrive rapidly and the available memory is typicallylimited. To meet the real-time requirement of data streams, everythingis preferably done in memory. Moreover, it should not incur a lot ofcomputation or storage overhead. However, the original onion index keepstrack of all tuples and, thus, requires a lot of storage space. Also, asnoted above, maintaining the original onion index is computationallyexpensive, making it difficult to meet the real-time requirement of datastreams. Therefore, the original onion index, as introduced in theabove-referenced “Onion Technique Article” does not work for datastreams.

SUMMARY

These and other drawbacks and disadvantages of the prior art areaddressed by the present principles, which are directed to a system andmethod for indexing a data stream.

According to an aspect of the present invention, there is provided amethod for indexing a data stream having attribute values. The methodincludes the steps of parsing the data stream and forming an index oftuples for a subset of attribute values of the data stream. The index isconfigured for retrieving the top-K tuples that optimize linearlyweighted sums of at least some of the attribute values in the subset.

According to another aspect of the present invention, there is provideda computer program product comprising a computer usable medium havingcomputer usable program code for indexing a data stream having attributevalues. The computer program product includes computer usable programcode for forming an index of tuples for a subset of attribute values ofthe data stream. The index is configured for retrieving the top-K tuplesthat optimize linearly weighted sums of at least some of the attributevalues in the subset.

According to yet another aspect of the present invention, there isprovided a system for indexing a data stream having attribute values.The system includes a data stream indexer for forming an index of tuplesfor a subset of attribute values of the data stream. The index isconfigured for retrieving the top-K tuples that optimize linearlyweighted sums of at least some of the attribute values in the subset.

These and other objects, features and advantages will become apparentfrom the following detailed description of illustrative embodimentsthereof, which is to be read in connection with the accompanyingdrawings.

BRIEF DESCRIPTION OF DRAWINGS

The disclosure will provide details in the following description ofpreferred embodiments with reference to the following figures wherein:

FIG. 1 is a diagram illustrating an onion index 100 in two-dimensionalspace, in accordance with the prior art;

FIG. 2 is a block diagram illustrating an exemplary networkedenvironment to which the present principles may be applied, according toan embodiment thereof;

FIG. 3 is a block diagram illustrating an exemplary computing device towhich the present principles may be applied, according to an embodimentthereof;

FIG. 4 is a diagram illustrating an exemplary data structure of an SAOindex, according to an embodiment of the present principles;

FIGS. 5A and 5B are diagrams illustrating an exemplary SAO index 500 intwo-dimensional space with approximate convex layer L₁ using up all ofthe storage budget, according to an embodiment of the presentprinciples;

FIG. 6 is a diagram illustrating the exemplary SAO index 500 of FIG. 5in two-dimensional space with the storage budget divided between twoapproximate convex layers, according to an embodiment of the presentprinciples;

FIG. 7 is a diagram illustrating the exemplary SAO index 500 of FIG. 6in two-dimensional space after tuple t expires, according to anembodiment of the present principles;

FIG. 8 is a flow diagram illustrating an exemplary method for uniformmemory allocation for an SAO index, according to an embodiment of thepresent principles;

FIG. 9 is a flow diagram illustrating an exemplary method for static,non-uniform memory allocation for an SAO index, according to anembodiment of the present principles;

FIG. 10 is a diagram illustrating a projection 1000 of tuples along thedirection of query q, according to an embodiment of the presentprinciples;

FIG. 11 is a flow diagram illustrating an exemplary method for dynamic,non-uniform storage allocation for an SAO index, according to anembodiment of the present principles;

FIG. 12 is a flow diagram illustrating an exemplary method for selectinga victim approximate convex layer with respect to the dynamic,non-uniform storage allocation method 1100 of FIG. 11, according to anembodiment of the present principles;

FIG. 13 is a flow diagram illustrating an exemplary method for selectinga victim tuple with respect to the dynamic, non-uniform storageallocation method 1100 of FIG. 11, according to an embodiment of thepresent principles;

FIG. 14 is diagram illustrating a top-K linear optimization query q towhich the present principles may be applied, according to an embodimentthereof;

FIG. 15 is a flow diagram illustrating an exemplary method formaintaining an SAO index, according to an embodiment of the presentprinciples;

FIG. 16 is a flow diagram illustrating an exemplary method for tupleinsertion with respect to the index maintenance method 1500 of FIG. 15,according to an embodiment of the present principles;

FIGS. 17A and 17B are diagrams illustrating an example of inserting anew tuple t into a SAO index 1700, according to an embodiment of thepresent principles;

FIG. 18 is a flow diagram illustrating an exemplary method for tupleexpiration with respect to the index maintenance method 1500 of FIG. 15,according to an embodiment of the present principles; and

FIGS. 19A and 19B are diagrams illustrating an example of deleting atuple t′ from a SAO index 1900 according to the present principles,according to an embodiment of the present principles.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Embodiments of the present principles are directed to systems andmethods for indexing a data stream.

For purposes of brevity and illustration, embodiments of the presentprinciples are described herein with respect to the use of maximizationqueries for indexing the data streams. However, it is to be appreciatedthat, given the teachings of the present principles provided herein,embodiments of the present principles may be readily employed for bothmaximization and minimization queries, as well as other data streamapplications, while maintaining the scope of the present invention.Accordingly, the phrase “optimizing” as used herein, shall refer to bothmaximizing and/or minimizing linearly weighted sums of attributes withrespect to an optimization query.

It should be understood that the elements shown in the FIGURES may beimplemented in various forms of hardware, software or combinationsthereof. Preferably, these elements are implemented in software on oneor more appropriately programmed general-purpose digital computershaving a processor and memory and input/output interfaces.

Embodiments of the present invention can take the form of an entirelyhardware embodiment, an entirely software embodiment or an embodimentincluding both hardware and software elements. In a preferredembodiment, the present invention is implemented in software, whichincludes but is not limited to firmware, resident software, microcode,etc.

Furthermore, the invention can take the form of a computer programproduct accessible from a computer-usable or computer-readable mediumproviding program code for use by or in connection with a computer orany instruction execution system. For the purposes of this description,a computer-usable or computer readable medium can be any apparatus thatmay include, store, communicate, propagate, or transport the program foruse by or in connection with the instruction execution system,apparatus, or device. The medium can be an electronic, magnetic,optical, electromagnetic, infrared, or semiconductor system (orapparatus or device) or a propagation medium. Examples of acomputer-readable medium include a semiconductor or solid state memory,magnetic tape, a removable computer diskette, a random access memory(RAM), a read-only memory (ROM), a rigid magnetic disk and an opticaldisk. Current examples of optical disks include compact disk—read onlymemory (CD-ROM), compact disk—read/write (CD-R/W) and DVD.

A data processing system suitable for storing and/or executing programcode may include at least one processor coupled directly or indirectlyto memory elements through a system bus. The memory elements can includelocal memory employed during actual execution of the program code, bulkstorage, and cache memories which provide temporary storage of at leastsome program code to reduce the number of times code is retrieved frombulk storage during execution. Input/output or I/O devices (includingbut not limited to keyboards, displays, pointing devices, etc.) may becoupled to the system either directly or through intervening I/Ocontrollers.

Network adapters may also be coupled to the system to enable the dataprocessing system to become coupled to other data processing systems orremote printers or storage devices through intervening private or publicnetworks. Modems, cable modem and Ethernet cards are just a few of thecurrently available types of network adapters.

Referring now to the drawings in which like numerals represent the sameor similar elements and initially to FIG. 2, an exemplary networkedenvironment to which the present principles may be applied, is indicatedgenerally by the reference numeral 200. The environment 200 includes oneor more client devices 210 connected to a server 220 via a network 230.The network 230 may include wired and/or wireless links. The server 220may be connected in signal communication with one or more resources 240.The resources 240 may include one or more local and/or remote sources.The resources 240 may be connected to the server 220 directly and/orvia, e.g., one or more networks 240 (including wired and/or wirelesslinks). Each of the client devices 210 may include a stream indexingsystem 299 for creating SAO indexes as described herein.

Turning to FIG. 3, an exemplary computing device to which the presentprinciples may be applied is indicated generally by the referencenumeral 300. It is to be appreciated that elements of the computingdevice 300 may be employed in any of the client devices 210, the server220, and/or the resources 240. Moreover, it is to be further appreciatedthat elements of the computing device 300 may be employed in the streamindexing system 299.

The computing device 100 includes at least one processor (CPU) 102operatively coupled to other components via a system bus 104. A readonly memory (ROM) 106, a random access memory (RAM) 108, a displayadapter 110, an I/O adapter 112, a user interface adapter 114, a soundadapter 199, and a network adapter 198, are operatively coupled to thesystem bus 104.

A display device 116 is operatively coupled to system bus 104 by displayadapter 110. A disk storage device (e.g., a magnetic or optical diskstorage device) 118 is operatively coupled to system bus 104 by I/Oadapter 112.

A mouse 120 and keyboard 122 are operatively coupled to system bus 104by user interface adapter 114. The mouse 120 and keyboard 122 are usedto input and output information to and from system 100.

At least one speaker (herein after “speaker”) 197 is operatively coupledto system bus 104 by sound adapter 199. A (digital and/or analog) modem196 is operatively coupled to system bus 104 by network adapter 198.

Advantageously, a Stream Approximate Onion-like structure (SAO) indexhaving a plurality of convex layers and corresponding vertices isdisclosed herein. The SAO index may be used to provide approximateanswers to arbitrary linear optimization queries almost instantaneously.In contrast to the Onion index of the prior art, embodiments of an SAOindex in accordance with the present principles may maintain only thefirst few outer convex layers. Moreover, in contrast to the Onion indexof the prior art, embodiments of an SAO index in accordance with thepresent principles may keep only some of the most “important” verticesin each layer in the SAO index rather than all vertices.

In some embodiments of the present principles, a dynamic, non-uniformstorage allocation strategy is used, such that a larger portion ofavailable memory tends to be allocated to the outer layers than to theinner layers. In this way, both storage and maintenance overheads of theSAO index are greatly reduced with respect to the Onion index of theprior art. Additionally, the errors introduced into the approximateanswers are also minimized.

The SAO index reduces both the index storage overhead and the indexmaintenance overhead in relation to the Onion index of the prior art, bykeeping only a subset of the tuples in a data stream in the SAO index.In an embodiment, a count-based sliding window model is used for datastreams, with W denoting the sliding window size. That is, the tuplesunder consideration are the last W tuples that have been viewed. Giventhe teachings of the present invention provided herein, one of ordinaryskill in this and related arts may readily extend an SAO index to thecase of time-based sliding windows or the case that the entire historyof the stream is considered, while maintaining the scope of the presentinvention.

Suppose the available memory can hold M+1 tuples. In the steady state,no more than M tuples are kept in the SAO index. That is, the storagebudget is M tuples. However, in a transition period, M+1 tuples can bekept in the SAO index temporarily. In general, a tuple includes bothfeature attributes and non-feature attributes. Even if the convex hulldata structures for feature attribute vectors occupy a small amount ofstorage space, the non-feature attributes may still dominate the storagerequirement. For example, in the environmental epidemiology applicationmentioned in the introduction, each tuple has a large non-feature imageattribute. For linear optimization queries, we are interested in findingall attributes of the top-K tuples. Hence, the exact value of M dependson the specific application. Given the teachings of the presentprinciples provided herein, the present principles can be readilyextended with some modification, to the case where the available memoryis measured in terms of bytes.

The SAO index carefully controls the number of tuples on each layer toprovide good approximate answers to linear optimization queries. Tofully utilize available memory as much as possible, embodiments of theSAO index may dynamically allocate the proper amount of storage toindividual layers, as described in detail herein below, so that a largerportion of the available memory tends to be allocated to the outerlayers. In this way, the quality of the approximate answers can bemaximized without increasing the memory requirement. In the case ofmemory overflow, embodiments of an SAO index in accordance with thepresent principles may keep the most “important” tuples and discard theless “important” ones. Moreover, to minimize the computation overhead,embodiments of an SAO index in accordance with the present principlesmay utilize optimized creation and maintenance algorithms.

A description will now be given regarding an Impossibility Theorem withrespect to linear optimization queries and embodiments of an SAO indexin accordance with the present principles.

Users submitting linear optimization queries against data streamsgenerally have to accept approximate answers. If W≦M, all W tuples inthe sliding window can be kept in memory. Then for any linearoptimization query, the exact answer can always be computed by checkingthe last W tuples. However, if W>M, which is common in practice, it isimpossible to keep the last W tuples in memory. Then, according to thefollowing theorem, for any linear optimization query, the return ofexact answers cannot always be guaranteed. Hence, users have to acceptapproximate answers.

Theorem 2: In the case that W>M, for any top-K linear optimizationquery, no algorithm exists such that for any data distribution, theexact top-K tuples can always be found by just using the M tuples inmemory.

Proof: We focus on the one-dimensional case (n=1) with K=1. The proofcan be easily extended to the general case where n≧1 and K≧1. Consider alinear optimization query whose weight w₁>0 (the case that w₁<0 issymmetric). Suppose all tuples arrive in such an order that theirfeature attribute values (a₁ ^(j)) are monotonically decreasing. Then itis easy to see that as the sliding window moves, no algorithm can alwaysuse the M tuples in memory to keep track of the tuple that is both“valid” and has the largest feature attribute value. That is, we cannotalways use the M tuples in memory to find the exact top-1 tuple.

Hereinafter, for purposes of illustration and brevity, the case of W>Mis described. In this case, it is impossible to keep the precise onionindex in memory and use the precise onion index to provide exact answersto linear optimization queries. Rather, the SAO index is provided, whichcan provide approximate answers to linear optimization queries almostinstantaneously.

An embodiment of an SAO index in accordance with the present principlesmay employ the following index organization.

One consideration in implementing the SAO index is based on thefollowing observation: An onion index typically includes a large numberof convex layers, but most inner layers are not needed for answering themajority of linear optimization queries. For example, as mentioned inabove with respect to the onion index, to answer a top-K linearoptimization query, at most the first K outer convex layers have to besearched. In contrast, embodiments of an SAO index in accordance withthe present principles may keep only the first few outer convex layersrather than all convex layers. More specifically, in an embodiment of anSAO index in accordance with the present principles, a user who createsthe SAO index may specify a number L. In such embodiment of the presentprinciples, the SAO index keeps only the first L outer convex layers.

Intuitively, if most linear optimization queries use a large K (e.g.,20), L could be smaller than that K (e.g., L=10). However, if mostlinear optimization queries use a very small K (e.g., 1), L should be alittle larger than that K (e.g., L=2). The reason is as follows. As willbe described below with respect to allocating a proper amount of memoryto each layer, when K is very small, embodiments of the SAO index mayinclude a few backup convex layers. This is to prevent the undesirablesituation that a few tuples on the first K outer convex layers expireand then large errors are introduced into the approximate answers tosome linear optimization queries. On the other hand, when K is large,for a top-K linear optimization query, it is likely that the top-Ktuples can be found on the first J outer convex layers, where J<K. Inthis case, if a few tuples on these J convex layers expire, the otherconvex layers can serve as backups automatically. Hence, L does not needto be larger than K.

Since M is limited, in some circumstances, the SAO index may not keepthe precise first L outer convex layers. For example, in the worst case,all W tuples in the sliding window may reside on the first convex layerrather than spread over multiple convex layers. Therefore, for each ofthe first L outer convex layers, the SAO index may only be able to keepsome of the most “important” tuples rather than all the tuples belongingto that layer. In other words, in some embodiments of the SAO index,each layer in the SAO index is an approximate convex layer in the sensethat it is an approximation to the corresponding precise convex layer inthe onion index. For each i (1≦i≦L), L_(i) is used to denote the ithapproximate convex layer.

Embodiments of the SAO index may maintain the following properties. Eachapproximate convex layer is the convex hull of all tuples on that layer.For each i (1≦i≦L−1), L_(i+1) is contained within L_(i). Also, the totalnumber of tuples on all L approximate convex layers is no more than M.Recall that as mentioned above, in a transition period of someembodiments, M+1 tuples may be kept in the SAO index temporarily.

In an embodiment, all the tuples in the SAO index may be kept as asorted doubly-linked list L_(dl). The sorting criterion may be a tuple'sremaining lifetime. Accordingly, the first tuple in L_(dl) is going toexpire the soonest. In this way, we can quickly check whether any tuplein the SAO index expires, which may be utilized for index maintenance asdescribed further below. Also, we can easily delete tuples that are inthe middle of L_(dl), which may be done when the available memory isexhausted and a tuple needs to be deleted from the SAO index.Corresponding considerations are further described below with respect todynamic, non-uniform storage allocation.

For each approximate convex layer, a standard convex hull data structuremay be maintained. The vertices of the convex hull point to tuples inL_(dl). Also, each tuple t in L_(dl) may have a label indicating theapproximate convex layer to which tuple t belongs. This label may beused when a tuple expires and the tuple needs to be removed from thecorresponding approximate convex layer. Corresponding considerations arefurther described below with respect to SAO index maintenance.

Embodiments of an SAO index in accordance with the present principlesmay employ various techniques for allocating the proper amount of memoryto each layer. Some of these various techniques will now be described.

It is to be appreciated that a consideration in implementing an SAOindex in accordance with the present principles is the approach to betaken to properly allocate memory to each layer, given a finite amountof memory, so that the quality of the approximate answers can bemaximized. Accordingly, examples of why memory allocation should be usedare provided, followed by three exemplary allocation strategies, namely:a simple, uniform strategy; a static, non-uniform strategy; and adynamic, non-uniform strategy. Of course, given the teachings of thepresent invention provided herein, one of ordinary skill in this andrelated arts will contemplate these and various other memory allocationstrategies for an SAO index in accordance with the present principles,while maintaining the scope of the present principles.

It is preferable that the SAO index controls the number of tuples oneach approximate convex layer. Otherwise, one or a few approximateconvex layers may use up all of the storage budget M. As a consequence,the SAO index may not provide good approximate answers to certain linearoptimization queries.

FIG. 4 is a diagram illustrating an exemplary data structure of an SAOindex according to the present principles. In FIG. 4, suppose L₁, thefirst approximate convex layer, uses up all of the storage budget M andall the other L−1 approximate convex layers are empty (It is to be notedthat the convention shown in the above-referenced “Onion TechniqueArticle” with respect to using dotted polygons to represent approximateconvex layers is also used herein). In this case, the information aboutall tuples inside L₁ is lost. These tuples are represented by the hollowcircles in FIGS. 5A and 5B and are, thus, called hollow tuples 510.Non-hollow tuples are represented by the filled-in circles 520. FIGS. 5Aand 5B are diagrams illustrating an exemplary SAO index 500 intwo-dimensional space with approximate convex layer L₁ using up all ofthe storage budget, according to the present principles. In particular,FIG. 5A illustrates an example corresponding to a time before a tuple texpires, and FIG. 5B illustrates an example corresponding to a timeafter tuple t has expired.

Consider a top-1 linear optimization query q whose direction isrepresented by the arrow in FIGS. 5A and 5B. When tuple t expires fromthe sliding window, the SAO index cannot provide good approximate answerto q. This is because the linear combination values of those hollowtuples are all much larger than the maximal linear combination value ofthe remaining tuples on L₁. However, those hollow tuples are not kept inthe SAO index.

Now suppose the SAO index controls the number of tuples on eachapproximate convex layer. For example, the storage budget M is dividedamong all L approximate convex layers in a more balanced way, as shownin FIG. 6. FIG. 6 is a diagram illustrating the exemplary SAO index 500of FIG. 5 in two-dimensional space with the storage budget dividedbetween two approximate convex layers in a balanced way, according tothe present principles. This has the effect that some of the informationcontained in L₁ is lost while some other information can be kept in theother L−1 approximate convex layers.

Then after tuple t expires, L₁ can be “recovered” by using theinformation contained in L₂ (an exemplary recovery procedure isdescribed herein below with respect to index maintenance), as shown inFIG. 7. FIG. 7 is a diagram illustrating the exemplary SAO index 500 ofFIG. 6 (with the storage budget M divided among all L approximate convexlayers) in two-dimensional space after tuple t expires, according to thepresent principles. As a result, the SAO index can still provide a goodapproximate answer to the linear optimization query q.

Embodiments of an SAO index in accordance with the present principlesmay utilize a simple, uniform storage allocation strategy.

Turning to FIG. 8, an exemplary method for uniform memory allocation foran SAO index is indicated generally by the reference numeral 800.

As shown in FIG. 8, a simple storage allocation strategy in accordancewith one embodiment of the present principles is to divide the storagebudget M evenly among all L approximate convex layers (step 810). In theembodiment, each approximate convex layer does not keep more than M/Ltuples.

However, this simple, uniform method may not always provide an optimalallocation of memory. The reason is as follows. In the precise onionindex, according to Property 1, for a linear optimization query, we tendto find more of the top-K tuples on the outer convex layers than on theinner convex layers. For example, consider a top-20 linear optimizationquery. The precise onion index may find the largest ten tuples on thefirst convex layer, the next largest six tuples on the second convexlayer, and the remaining largest four tuples on the third convex layer.This is consistent with an observation made of the onion index of theabove-referenced “Onion Technique Article”, namely: to retrieve thetop-K tuples, typically we only need to access a few outer convex layersrather than all first K outer convex layers in the precise onion index.

The approximating SAO index is implemented so as to have a similarproperty with respect to the precise onion index, namely: for a linearoptimization query, we tend to find more of the top-K tuples on theouter approximate convex layers than on the inner approximate convexlayers. Intuitively, the more tuples allocated to an approximate convexlayer L_(i) (1≦i≦L) in the SAO index, the closer L_(i) is to thecorresponding precise convex layer and, thus, the more precise the toptuples we find on L_(i). Moreover, as discussed below, compared to thetop tuples that are found on the inner approximate convex layers, thetop tuples that are found on the outer approximate convex layers areranked higher and, thus, more important. Therefore, to provide goodapproximate answers to linear optimization queries, the SAO index shouldallocate more tuples to the outer approximate convex layers than to theinner approximate convex layers.

Embodiments of an SAO index in accordance with the present principlesmay employ a static, non-uniform storage allocation strategy.

Turning to FIG. 9, an exemplary method for static, non-uniform memoryallocation for an SAO index is indicated generally by the referencenumeral 900.

In an embodiment corresponding to the case that resources are limited,we determine the optimal numbers of tuples the SAO index should allocateto the L approximate convex layers. As used herein, the phrase“resources are limited” refers to the condition where each approximateconvex layer needs more tuples than can be actually allocated to it.Hereinafter, other embodiments of an SAO index are described thatutilize a dynamic, non-uniform storage allocation strategy that is basedon the results derived with respect to the static, non-uniform storageallocation strategy.

In the case that resources are limited, for each i (1≦i≦L), let N_(i)denote the optimal number of tuples that should be allocated to L_(i).Then, N_(i) is determined as follows:

$\begin{matrix}{{\sum\limits_{i = 1}^{L}N_{i}} = {M.}} & (1)\end{matrix}$

In general, the values of N_(i)'s depend on the exact data distribution.Since the data distribution is usually not known in advance, N_(i)'scannot be determined exactly. In our derivation, a few simplifiedpresumptions are made. This makes our derived N_(i)'s heuristic innature rather than exactly optimal.

Consider a top-L linear optimization query. For each i (1≦i≦L), lett_(i) represent the exact ith largest tuple, and t_(i)′ represent theith largest tuple that is found in the SAO index. Here, v_(i) is thelinear combination value of t_(i), and v_(i)′ is the linear combinationvalue of t_(i)′. The relative error of t_(i)′ is defined as follows:

$\begin{matrix}{e_{i} = {{\frac{v_{i} - v_{i}^{\prime}}{v_{i}}}.}} & (2)\end{matrix}$

For the top-L tuples (t_(i)′) that are returned by the SAO index, aweighted mean of their relative errors is used as the performance metrice:

$\begin{matrix}{{e = \frac{\sum\limits_{i = 1}^{L}{u_{i}e_{i}}}{\sum\limits_{i = 1}^{L}u_{i}}},} & (3)\end{matrix}$where u_(i) is the weight of e_(i). Intuitively, the higher the rank ofa tuple t, the more important t's relative error. Hence, u_(i) should bea non-increasing function of i. We would like to minimize the mean of efor all top-L linear optimization queries. This is the condition basedon which N_(i)'s are derived.

Let p_(ij) (1≦i≦L, 1≦j≦L) represent the probability that for a top-Llinear optimization query, tuple t_(i) is on the jth convex layer in theonion index. We assume that in this case, tuple t_(i)′ is also on L_(j),the jth approximate convex layer in the SAO index. Furthermore, the meanof e_(i) for all top-L linear optimization queries is 1/N_(j), based onthe intuition that the larger the N_(j), the closer L_(j) is to the jthprecise convex layer in the onion index and, thus, the smaller thee_(i).

For illustrative purposes, a heuristic justification for the assumptionof 1/N_(j) is provided as follows. We project all N_(j) tuples on L_(j)along the query direction, as shown in FIG. 10, which is a diagramillustrating a projection 1000 of tuples along the direction of query qin accordance with an exemplary embodiment of the present principles.Each projection is a point. For these N_(j) points, let d denote theaverage distance between two adjacent points. Presume that on average,half of the N_(j) points are to the left of the origin, and the otherhalf of the N_(j) points are to the right of the origin. The projectionof tuple t_(i)′ is point V′_(i). We have E(v′_(i))=dN_(j)/2, where E(x)represents the expectation of x. Note that v_(i), the projection oftuple t_(i), is to the right of point v_(i)′. Suppose the averagedistance between v_(i)′ and v_(i) is d/2. Then for a specific j, themean of e_(i) is as follows:

${{E( {\frac{v_{i} - v_{i}^{\prime}}{v_{i}}} )} \approx {E( {\frac{v_{i} - v_{i}^{\prime}}{v_{i}}} )} \approx \frac{E( {{v_{i} - v_{i}^{\prime}}} )}{E( {v_{i}^{\prime}} )}} = {{\frac{d/2}{d\;{N_{j}/2}}} = {\frac{1}{N_{j}}.}}$

Now we return to the goal of minimizing ē, the mean of e. For each i(1≦i≦L), tuple t_(i) must be on one of the L convex layers in the onionindex. Hence, ē_(i), the mean of e_(i), is a weighted average over allj's (l≦j≦L) as follows:

${\overset{\_}{e}}_{i} = {\sum\limits_{j = 1}^{L}{p_{ij}{\frac{1}{N_{j}}.}}}$

From Equation (3), we have the following:

${\overset{\_}{e}{\sum\limits_{i = 1}^{L}u_{i}}} = {{\sum\limits_{i = 1}^{L}{u_{i}{\overset{\_}{e}}_{i}}} = {{\sum\limits_{i = 1}^{L}( {u_{i}{\sum\limits_{j = 1}^{L}{p_{ij}\frac{1}{N_{j}}}}} )} = {\sum\limits_{j = 1}^{L}{( {\frac{1}{N_{j}}{\sum\limits_{i = 1}^{L}{u_{i}p_{ij}}}} ).}}}}$

Define C_(j) as follows:

$\begin{matrix}{C_{j} = {\sum\limits_{i = 1}^{L}{u_{i}{p_{ij}.}}}} & (4)\end{matrix}$

We have the following:

$\begin{matrix}{{\overset{\_}{e}{\sum\limits_{i = 1}^{L}u_{i}}} = {{\sum\limits_{j = 1}^{L}{\frac{1}{N_{j}}C_{j}}} = {\sum\limits_{j = 1}^{L}{\sqrt{C_{j}}{\frac{\sqrt{C_{j}}}{N_{j}}.}}}}} & (5)\end{matrix}$

From Equation (1), we obtain the following:

$\begin{matrix}{M = {{\sum\limits_{j = 1}^{L}N_{j}} = {\sum\limits_{j = 1}^{L}{\sqrt{C_{j}}{\frac{N_{j}}{\sqrt{C_{j}}}.}}}}} & (6)\end{matrix}$

To minimize ē, the following weighted arithmetic-harmonic meansinequality is used:

Theorem 3. Given L positive weights w₁, w₂, . . . , w_(L) and L positivenumbers x₁, x₂, . . . , x_(L), we have weighted arithmetic mean≧2weighted harmonic mean, with equality only when x₁=x₂= . . . =x_(L).That is,

$\begin{matrix}{\frac{\sum\limits_{j = 1}^{L}{w_{j}x_{j}}}{\sum\limits_{j = 1}^{L}w_{j}} \geq {\frac{\sum\limits_{j = 1}^{L}w_{j}}{\sum\limits_{j = 1}^{L}{w_{j}\frac{1}{x_{j}}}}.}} & (7)\end{matrix}$

After transforming (7), we have

$\begin{matrix}{{\sum\limits_{j = 1}^{L}{w_{j}\frac{1}{x_{j}}}} \geq {\frac{( {\sum\limits_{j = 1}^{L}w_{j}} )^{2}}{\sum\limits_{j = 1}^{L}{w_{j}x_{j}}}.}} & (8)\end{matrix}$

Let

$\begin{matrix}{w_{j} = \sqrt{C_{j}}} & {and} & {x_{j} = {\frac{N_{j}}{\sqrt{C_{j}}}.}}\end{matrix}$Using Equations (5), (6), and (8), we know that ē (or alternatively, theleft side of Equation (5)) is minimized when the following conditionholds:

$\begin{matrix}{\frac{N_{1}}{\sqrt{C_{1}}} = {\frac{N_{2}}{\sqrt{C_{2}}} = {\ldots = {\frac{N_{L}}{\sqrt{C_{L}}}.}}}} & (9)\end{matrix}$

Then from Equation (1), we obtain the following:

$\begin{matrix}{N_{j} = {\frac{\sqrt{C_{j}}}{\sum\limits_{i = 1}^{L}\sqrt{C_{i}}}{M.}}} & (10)\end{matrix}$

According to Property 1, we know that if i<j, p_(ij)=0. If we assumethat t_(i), the exact ith largest tuple, has equal probability to be onany one of the first i outer convex layers in the onion index, then wehave the following:

$p_{ij} = \{ {\begin{matrix}{1/i} & ( {i \geq j} ) \\0 & ( {i < j} )\end{matrix}.} $

In accordance with the present principles, for illustration purposes, wepick u_(i)=1/i. It is to be appreciated that other choices of u_(i) canbe used, while maintaining the scope of the present principles. Theresults are similar and, thus, omitted herein, but are readilydetermined by one of ordinary skill in this and related arts. Then, fromEquation (4), we have the following:

$C_{j} = {{\sum\limits_{i = j}^{L}{u_{i}p_{ij}}} = {\sum\limits_{i = j}^{L}{\frac{1}{i^{2}}.}}}$

Turning to FIG. 9, an exemplary method for static, non-uniform memoryallocation for an SAO index is indicated generally by the referencenumeral 900. For purposes of illustration and brevity, the method 900 isdescribed in a cursory manner, having been described in detail hereinabove.

In determining the optimal number N_(j) of tuples that should beallocated to layer L_(j), first a variable C_(j) is calculated asfollows:

$\begin{matrix}{C_{j} = {{\sum\limits_{i = j}^{L}{u_{i}p_{ij}}} = {\sum\limits_{i = j}^{L}\frac{1}{i^{2}}}}} & ( {{step}\mspace{14mu} 910} )\end{matrix}$

Then, using the value of C_(j) calculated at step 910, N_(j) iscalculated as follows:

$\begin{matrix}{N_{j} = {\frac{\sqrt{C_{j}}}{\sum\limits_{i = 1}^{L}\sqrt{C_{i}}}M}} & ( {{step}\mspace{14mu} 920} )\end{matrix}$

It is to be appreciated that steps 910 and 920 may be performed for eachlayer of the SAO index. In this way, an optimal memory allocation may beimplemented for each layer of the SAO index.

Embodiments of an SAO index in accordance with the present principlesmay utilize dynamic, non-uniform storage allocation.

In an embodiment, if for each i (1≦i≦L), L_(i), the ith approximateconvex layer always needs more than N_(i) tuples, then the SAO index canuse a static storage allocation strategy so that L_(i) gets a fixedstorage quota of N_(i) tuples. However, real world application mayrequire more of a dynamic response. At any time, some approximate convexlayers may need more than N_(i) tuples while other approximate convexlayers may need fewer than N_(i) tuples. As tuples keep entering andleaving the sliding window, the storage requirements of differentapproximate convex layers change continuously. If the SAO index stickswith the static storage allocation strategy, the total storage quota ofM tuples cannot always be fully utilized. For example, this is the caseif some approximate convex layers do not use up their storage quotaN_(i). This will hurt the quality of the approximate answers the SAOindex provides to linear optimization queries.

To ensure the best quality of the approximate answers that are providedto linear optimization queries, the SAO index needs to fully utilize thestorage budget M as much as possible. Therefore, instead of using thestatic storage allocation strategy, the SAO index does dynamic storageallocation. In this way, the approximate convex layers that need extrastorage quota can “borrow” some quota from those approximate convexlayers that have spare quota.

With respect to dynamic, non-uniform storage allocation that may be usedby some embodiments of an SAO index in accordance with the presentprinciples, our design principle is that whenever possible, the storagebudget M is used up. At the same time, the SAO index maintains condition(9) as much as possible. That is, the number of tuples on L_(i) isproportional to √{square root over (C_(i))}.

In accordance with one embodiment, an exemplary method for dynamic,non-uniform storage allocation for an SAO index is indicated generallyby the reference numeral 1100 in FIG. 11.

The dynamic, non-uniform storage allocation method 1100 is described asfollows. For each i (1≦i≦L), let M_(i) denote the number of tuples onL_(i) (step 1110). The SAO index continuously monitors these M_(i)'s(step 1120). At any time, there are two possible cases. In the firstcase,

${{\sum\limits_{i = 1}^{L}M_{i}} \leq M},$which is determined at step 1130. This is the safe case and nothingneeds to be done, as the storage budget M has not been used up. In thesecond case,

${{\sum\limits_{i = 1}^{L}M_{i}} = {M + 1}},$which is determined at step 1140. According to our SAO index maintenancestrategy that is described herein below,

$\sum\limits_{i = 1}^{L}M_{i}$can never be larger than M+1. This is the unsafe case, as the storagebudget M is exceeded by one. Thus, a victim approximate convex layer hasto be selected and one tuple has to be deleted from the selected layer(step 1150).

Note that the dynamic storage allocation strategy is of a finegranularity. Each time memory is exhausted, one tuple is deleted fromthe SAO index. One may consider whether we could use a dynamic storageallocation strategy that is of a coarser granularity. That is, each timememory is exhausted, multiple tuples (rather than a single tuple) aredeleted from the SAO index. Then, it will take longer before memory isexhausted again. However, such a method is not desirable in ourenvironment. This is because our storage budget is precious, as M may besmall. We want to fully utilize the limited storage budget as much aspossible so that the SAO index can provide the best approximate answersto linear optimization queries. Moreover, as can be seen from thedescription of step 1 of the exemplary index maintenance describedherein below, the insertion of a new tuple into the SAO index may causemultiple tuples to be expelled from L_(L) and then some storage budgetbecomes available automatically.

In an embodiment relating to a dynamic, non-uniform storage strategy, avictim approximate convex layer may be chosen as follows. It is to beappreciated that the following methodology is merely illustrative and,given the teachings of the present invention provided herein, one ofordinary skill in this and related arts may contemplate this and othermethodologies for choosing a victim approximate convex layer, whilemaintaining the scope of the present invention.

Turning to FIG. 12, an exemplary method for selecting a victimapproximate convex layer with respect to the dynamic, non-uniformstorage allocation method 1100 of FIG. 11 is indicated generally by thereference numeral 1200. For each i (1≦i≦L), let r_(i)=M_(i)/N_(i) (step1210). We pick j such that r_(j)=max{r_(i)|r_(i)>1,1≦i≦L} (step 1220).This j must exist. Otherwise for each i (1≦i≦L), r_(i)≦1. This leads to

${{{\sum\limits_{i = 1}^{L}M_{i}} \leq {\sum\limits_{i = 1}^{L}N_{i}}} = M},$which conflicts with the condition that

${\sum\limits_{i = 1}^{L}M_{i}} = {M + 1.}$L_(j) is chosen as the victim approximate convex layer (step 1230).

The above method is based on the intuition that the victim approximateconvex layer L_(j) should satisfy the following two conditions. First,L_(j) has used up its fixed quota N_(j). Second, among all approximateconvex layers that have used up their fixed quota N_(i), L_(j) exceedsits fixed quota (by the ratio r_(j)) the most. In this way, we can befair to those approximate convex layers that have not used up theirfixed quota N_(i). Also, the SAO index can maintain the conditionM_(i)∝√{square root over (C_(i))} as much as possible.

In an embodiment relating to a dynamic, non-uniform storage strategy, avictim tuple may be chosen as follows. It is to be appreciated that thefollowing methodology is merely illustrative and, given the teachings ofthe present invention provided herein, one of ordinary skill in this andrelated arts may contemplate this and other methodologies for choosing avictim tuple, while maintaining the scope of the present invention.

Turning to FIG. 13, an exemplary method for selecting a victim tuplewith respect to the dynamic, non-uniform storage allocation method 1100of FIG. 11 is indicated generally by the reference numeral 1300.

Now one victim tuple needs to be deleted from the victim approximateconvex layer L_(j). Intuitively, this victim tuple t should have a closeneighbor so that deleting t will have little impact on the shape ofL_(j). Two tuples on an approximate convex layer are neighbors if theyare connected by an edge.

For any tuple t on L_(j), let R_(t) denote the Euclidean distancebetween tuple t and its nearest neighbor on L_(j) (step 1310). Thevictim tuple is chosen to be the tuple that has the smallest R_(t)(usually there are two such tuples and the older one, i.e., thesooner-to-expire one, is picked) (step 1320). Note that R_(t) is not thesmallest distance between tuple t and any other tuple on L_(j). Rather,in computing R_(t), only tuple t's neighbors are considered.

For illustrative purposes, an example will now be provided to illustratethe reasoning. Consider the victim approximate convex layer L_(j) of anSAO index in two-dimension space according to an exemplary embodiment ofthe present principles, shown in FIG. 14. If R_(t) denotes the smallestdistance between tuple t and any other tuple on L_(j), then tuples t₁and t₃ have the smallest R_(t). Suppose t₃ is older than t₁. In thiscase, t₃ is picked as the victim tuple and deleted from L_(j). Thisgreatly influences the shape of L_(j). There are two possible cases, andproblems may possibly be encountered in either case.

In the first case, j=L. Turning to FIG. 14, a top-K linear optimizationquery q (having a direction as shown) is indicated generally by thereference numeral 1400. Suppose the Kth largest tuple of q comes fromL_(L). Then, the SAO index cannot provide a good answer for the Kthlargest tuple of q, since the information about all tuples inside L_(L)is lost.

In the second case, j<L. Due to the dramatic shape change of L_(j), itis likely that after deleting t₃, L_(j) will overlap with L_(j+1). Inthis second case, as will be described herein below with respect to theindex maintenance, the SAO index needs to adjust L_(j+1) and maybe someapproximate convex layers inside L_(j+1). This is rather time-consuming.

In contrast, if R_(t) denotes the distance between tuple t and itsnearest neighbor on L_(j), then tuples t₁ and t₂ have the smallestR_(t). Irrespective of whether t₁ or t₂ is deleted from L_(j), there isonly a minor change to the shape of L_(j) and, thus, we are not likelyto run into the trouble described above.

With respect to deleting a victim tuple, after choosing the victim tuplet, we may use the method that is described with respect to step 1520 ofFIG. 15 herein below with respect to index maintenance, to delete t fromL_(j) and then adjust the affected approximate convex layers.

A description will now be given regarding index creation for anembodiment of an SAO index in accordance with the present principles. Atthe beginning, the SAO index is empty. We keep receiving new tuplesuntil there are M tuples. Then, a standard convex hull constructionalgorithm, such as the quickhull method, may be used to create the Lapproximate convex layers in a batch. This is mainly for efficiencypurposes, as creating convex hulls in batch is less expensive thanconstructing convex hulls incrementally (i.e., each time adding one newtuple). Note that it is possible that some of the innermost approximateconvex layers are empty. Of course, it is to be appreciated that otherconvex hull construction methods may also be employed to construct anSAO index in accordance with the present principles, while maintainingthe scope of the present principles.

From now on, each time a new tuple arrives, we use the method describedherein below with respect to index maintenance to incrementally maintainthe SAO index. It is to be appreciated that the embodiment of thepresent principles are not limited to the preceding methodologies withrespect to index creation and, thus, other methodologies may also beemployed while maintaining the scope of the present principles.

Embodiments of an SAO index in accordance with the present principlesmay employ index maintenance. It is to be appreciated that the followingembodiments relating to index maintenance are merely illustrative and,given the teachings of the present principles provided herein, one ofordinary skill in this and related arts may utilize the following andother methodologies for index maintenance for an SAO index, whilemaintaining the scope of the present invention.

In a typical data streaming environment, we expect that W>>M. That is,only a small fraction of all W tuples in the sliding window are storedin the SAO index. Intuitively, this means that tuples on the approximateconvex layers can be regarded as anomalies. The smaller the i (1≦i≦L),the more anomalous the tuples on L_(i). As a result, we have thefollowing heuristic (not exact) property:

Property 2: Most new tuples are “normal” tuples and thus inside L_(L).Moreover, for a new tuple t, it is most likely to be inside L_(L). Lesslikely is tuple t between L_(L−1) and L_(L), and even less likely istuple t between L_(L−2) and L_(L−1), etc.

According to our exemplary storage allocation strategies describedherein above, the inner approximate convex layers tend to have fewertuples than the outer approximate convex layers. From computationalgeometry literature, it is known that given a point p, the complexity ofchecking whether p is inside a convex polytope P increases with thenumber of vertices of P. Therefore, we have the following property:

Property 3: For a tuple t, it is typically faster to check whether t isinside an inner approximate convex layer than to check whether t isinside an outer approximate convex layer.

Turning to FIG. 15, an exemplary method for maintaining an SAO index isindicated generally by the reference numeral 1500.

Upon the arrival of a new tuple t, Properties 2 and 3 may be used toreduce the SAO index maintenance overhead. We may proceed in thefollowing exemplary three steps. Step 1510 checks whether tuple t shouldbe inserted into the SAO index. Step 1520 checks whether any tuple inthe SAO index expires. Step 1530 handles memory overflow.

A description of an illustrative embodiment of step 1510 of FIG. 15relating to tuple insertion for the purpose of index maintenance willnow be described with respect to the present principles.

Turning to FIG. 16, an exemplary method for tuple insertion with respectto the index maintenance method 1500 of FIG. 15 is indicated generallyby the reference numeral 1600.

Let set S={t} (step 1610). Set i=k (step 1620). It is then determinedwhether or not |S|>0 && i≦L (step 1630). If not, then the method isterminated. Otherwise, the expelled tuples are inserted into the currentlayer such that S=S∪{tuples on L_(i)} (step 1640). A new convex hull isconstructed such that L_(i)=convex hull of S (step 1650). The expelledtuples are obtained from L_(i) such that S=S−{tuples on L_(i)} (step1660). The method increments i (i++) to proceed to the next layer (step1670), and then returns to step 1630.

All approximate convex layers are checked one by one, starting fromL_(L). That is, our checking direction is from the innermost approximateconvex layer to the outermost approximate convex layer. From Properties2 and 3 together with the procedure described below, it can be seen thatthis checking direction is the most efficient one.

There are two possible cases. In the first case, tuple t is insideL_(L). According to Property 2, this is the mostly likely case. Also,according to Property 3, it can be discovered quickly whether tuple t isinside L_(L). In this first case, tuple t will not change any of the Lapproximate convex layers and thus can be thrown away immediately. Sinceno new tuple is introduced into the SAO index, there will be no memoryoverflow and, thus, Step 1530 can be skipped, although Step 1520 stillneeds to be performed. Note: If L_(L) is empty, we may consider thattuple t is outside of L_(L).

In the second case, a number k (1≦k≦L) can be located such that tuple tis inside L_(k−1) but outside of L_(k). If k=1, then tuple t is outsideof all L approximate convex layers. In this case, tuple t should beinserted into the SAO index. This insertion will affect L_(k) and maybesome approximate convex layers inside L_(k). However, none of the firstk−1 approximate convex layers will be affected.

This insertion is done in the following way. The new L_(k) is computedby considering both tuple t and all tuples on the existing L_(k), usingany standard incremental convex hull maintenance algorithm such as thebeneath-beyond method. This may cause one or more tuples to be expelledfrom L_(k). If that happens, the expelled tuples need to be furtherinserted into the next layer L_(k+1). In other words, the new L_(k+1) iscomputed by considering both the expelled tuples and all tuples on theexisting L_(k+1). This may again expel some tuples from L_(k+1). Theiteration continues until either L_(L) is reached or no more tuples areexpelled.

An exemplary embodiment of the insertion procedure is described below inpseudo code:

 Let set S = {t};  i = k;  while (|S|>0 && i≦L) {  S = S ∪ {tuples onL_(i)}; // insert expelled tuples into the current layer  L_(i) = convexhull of S; // construct a new convex hull  S = S − {tuples on L_(i)}; //obtain expelled tuples from L_(i)  i++; // go to the next layer  }

FIGS. 17A and 17B are diagrams illustrating an example of inserting anew tuple t into a SAO index 1700. In particular, FIG. 17A relates to atime before inserting a tuple t, and FIG. 17B relates to a time afterinserting a tuple t.

A description of an illustrative embodiment of step 1520 of FIG. 15.relating to tuple expiration for the purpose of index maintenance willnow be described with respect to the present principles.

Turning to FIG. 18, an exemplary method for tuple expiration withrespect to the index maintenance method 1500 of FIG. 15 is indicatedgenerally by the reference numeral 1800.

Let set S={t′} (step 1810). Set i=k (step 1820). It is then determinedwhether or not |S|>0 && i≦L (step 1830). If not, then the method isterminated. Otherwise, the remaining tuples on the current layer areobtained such that S₁={tuples on L_(i)}−S (step 1840). The remainingtuples are merged with the next layer such that S₂=S₁∪{tuples onL_(i+1)} (step 1850). A new convex hull is constructed such thatL_(i)=convex hull of S₂ (step 1860). Tuples from a lower layer areobtained for moving up to the current layer such that S={tuples onL_(i)}−S₁ (step 1870). The method increments i (i++) to proceed to thenext layer (step 1880), and then returns to step 1830.

The arrival of tuple t will cause at most one tuple in the SAO index toexpire from the sliding window. Let t′ denote the first tuple in thedoubly-linked list L_(dl). Recall that all tuples in L_(dl) are sortedin ascending order of their remaining lifetimes. Hence, only tuple t′needs to be checked, as t′ is the only tuple in the SAO index that mayexpire from the sliding window.

There are two possible cases. In the first case, tuple t′ has notexpired. We proceed to Step 1530 directly.

In the second case, tuple t′ has expired and thus needs to be deletedfrom the SAO index. Suppose tuple t′ is on L_(k) (1≦k≦L). The deletionof tuple t′ will affect L_(k) and maybe some approximate convex layersinside L_(k). However, none of the first k−1 approximate convex layerswill be affected.

This deletion is implemented as follows. The new L_(k) is computed byconsidering both all tuples on the existing L_(k) (except for tuple t′)and all tuples on L_(k+1). If one or more tuples on L_(k+1) are moved upto the new L_(k), then the new L_(k+1) needs to be further computed byconsidering both the remaining tuples on L_(k+1) and all tuples onL_(k+2). The iteration continues until either L_(L) is reached or nomore tuples are moved up. Since this iteration procedure reduces thenumber of tuples in the SAO index by one, there will be no memoryoverflow and, thus, Step 1530 can be skipped.

The deletion procedure is described herein after in pseudo code.

 Let set S = {t′};  i = k;  while (|S|>0 && i≦L) {  S₁ = {tuples onL_(i)} − S; // obtain remaining tuples on the current layer  S₂ = S₁ ∪{tuples on L_(i+1)}; // merge with the next layer  L_(i) = convex hullof S₂; // construct a new convex hull  S = {tuples on L_(i)} − S₁; //obtain tuples that are moved up to the current layer  i++; // go to thenext layer  }

FIGS. 19A and 19B are diagrams illustrating an example of deleting atuple t′ from a SAO index 1900 according to the present principles. Inparticular, FIG. 19A illustrates an example corresponding to a timebefore a tuple t is deleted, and FIG. 19B illustrates an examplecorresponding to a time after tuple t has been deleted.

A description of an illustrative embodiment of step 1530 of FIG. 15relating to the handing of memory overflow will now be described withrespect to the present principles.

In the above steps 1510 and 1520, at most one new tuple is introducedinto the SAO index while one or more tuples may be deleted (e.g., tuplesmay get expelled from L_(L) in Step 1510). Now we check whether or notthe condition

${\sum\limits_{i = 1}^{L}M_{i}} \leq M$still holds. Recall that M_(i) denotes the number of tuples on L_(i). Ifnot,

${\sum\limits_{i = 1}^{L}M_{i}} = {M + 1}$must be true. In this case, we use the procedure that is described abovewith respect to index maintenance to delete one tuple from the SAOindex.

From the above description, we can see that it may be computationallyexpensive to either insert a new tuple into the SAO index or delete anexisting tuple from the SAO index, as multiple approximate convex layersmay need to be reconstructed. Fortunately, upon the arrival of a newtuple, the amortized overhead of maintaining the SAO index is not thathigh. The reason is as follows.

First, according to Property 2, in most cases, the new tuple will beinside the innermost approximate convex layer L_(L) and, thus, can bethrown away immediately. Also, the number of tuples in the SAO index isat most M+1, which is usually much smaller than the sliding window sizeW. On average, after approximately W/M new tuples are received, onetuple in the SAO index expires. Therefore, we rarely need to eitherinsert a new tuple into the SAO index or delete a tuple from the SAOindex.

Second, M is typically not very large. Then for each i (1≦i≦L), M_(i),the number of tuples on L_(i), is also not very large. This reduces thereconstruction overhead of approximate convex layers, and also theoverhead of checking whether the new tuple is inside an approximateconvex layer.

Third, the SAO index maintenance algorithm in accordance with thepresent principles has been optimized. For example, an efficientchecking direction is used in Step 1510.

Embodiments of an SAO index in accordance with the present principlesmay employ a query evaluation procedure. It is to be appreciated thatthe following embodiments relating to query evaluation are merelyillustrative and, given the teachings of the present principles providedherein, one of ordinary skill in this and related arts may utilize thefollowing and other methodologies for query evaluation for an SAO index,while maintaining the scope of the present invention.

To provide approximate answers to a top-K linear optimization query (Kcan be larger than L), in some embodiments, we may employ the onionindex search procedure described in the above-referenced “OnionTechnique Article”. We start from L₁ and search the approximate convexlayers one by one. This search terminates when one of the following twoconditions are satisfied: (1) all L approximate convex layers have beensearched (in this case, all L approximate convex layers are treated asprevious approximate convex layers); or (2) the Kth largest tuple on theprevious approximate convex layers has larger linear combination valuethan the largest tuple on the current approximate convex layer. Then thetop-K tuples on the previous approximate convex layers are returned tothe user. According to Theorem 1, these K tuples are the top-K tuples inthe SAO index.

Embodiments of an SAO index in accordance with the present principlesmay be implemented in a parallel processing environment. The abovediscussion assumes that there is only one computer. If tuples arrive sorapidly that one computer cannot handle all of them, multiple (e.g., C)computers can be used. An illustrative embodiment involving parallelprocessing may be implemented as follows. All tuples are partitionedinto C sets (e.g., using round-robin partitioning). Each computermaintains a SAO index and handles a different set of tuples. When theuser submits a top-K linear optimization query, the local top-K tuplesare obtained on each computer. All these local top-K tuples are mergedtogether to get the global top-K tuples. This is our answer to the top-Klinear optimization query.

Having described preferred embodiments of a system and method (which areintended to be illustrative and not limiting), it is noted thatmodifications and variations can be made by persons skilled in the artin light of the above teachings. It is therefore to be understood thatchanges may be made in the particular embodiments disclosed which arewithin the scope and spirit of the invention as outlined by the appendedclaims. Having thus described aspects of the invention, with the detailsand particularity required by the patent laws, what is claimed anddesired protected by Letters Patent is set forth in the appended claims.

1. A method for indexing a data stream having attribute values,comprising: storing the data stream in a hardware memory device; parsingthe data stream in the hardware memory device; forming a StreamApproximate Onion (SAO) index of tuples for a subset of attribute valuesof the data stream, the index being configured for retrieving top-Ktuples that optimize linearly weighted sums of at least some of theattribute values in the subset; and configuring the index to track asubset of the top-K tuples in a sliding window applied to the datastream; wherein said forming step comprises forming the index to have aplurality of layers of convex hulls, and the method further comprises:maintaining in the index a number of the plurality of layers of convexhulls less than a number of convex hull layers maintained in an Onionindex; storing the index in the hardware memory device or anotherhardware memory device; and providing a user-discernable output based onthe index in response to an optimization query directed to the datastream, the user discernable output including the top-K tuples thatoptimize the linearly weighted sums of at least some of the attributevalues in the subset with respect to the optimization query.